The main difficulty in obtaining homeomorphisms on pseudo-arc with finite non-zero entropy is sufficient control of their dynamics. Theorem REF  serves as our starting point for constructing a more complicated homeomorphism with richer dynamics and to obtain the main result it is important that such starting homeomorphism carries no topological entropy. A very useful procedure that helps us in this direction is called the Denjoy-Rees technique and was described in [1]}. There, the authors used the technique to get the following result: any compact manifold of dimension at least two admitting a minimal uniquely ergodic homeomorphism also admits a uniquely ergodic homeomorphism with positive topological entropy.
The Denjoy-Rees technique can be viewed as an ingenious generalization of the famous Denjoy example [2]} (a periodic point free homeomorphism of the unit circle which is not conjugate to a rotation) and subsequently more involved Rees' construction [3]} (homeomorphism of the two-torus which is minimal and has positive topological entropy) with the additional new technique which allows to control how rich the dynamics of these homeomorphisms is.
We follow the Denjoy-Rees technique described above, however, several new difficulties arise in our setting. The first of them is that we are working on a one-dimensional space, unlike in [1]}. We solve this by providing an inverse limit version of the Denjoy-Ress technique; namely we construct our pseudo-arc as the inverse limit of pseudo-arcs and we are applying the technique on the coordinate spaces of the inverse limit. The other difficulty is that, after this enrichment, we want to get back to the space homeomorphic to the original one-dimensional space, the pseudo-arc. Here we use several properties of the pseudo-arc, namely it is important that the space is homogeneous so that we can find homeomorphism between neighbourhoods of different sets and points.
Also, crookedness of the pseudo-arc is used in a sense that the deformations we perform on the space throughout our construction cannot make it even more crooked. The argument would fail in the case of many other arc-like continua (e.g. for the arc or the Knaster continuum). In [1]} the authors state a collection of conditions \(\mathbf {A_1}\) -\(\mathbf {A_3}\) , \(\mathbf {B_1}\) -\(\mathbf {B_6}\)  and \(\mathbf {C_1}\) -\(\mathbf {C_8}\)  which allow them to obtain their main result. They state the conditions for rectangles which are in their context closed unit balls in \(\mathbb {R}^d\)  where \(d>1\)  is the dimension of a given compact topological manifold. For us, the rectangles from their setting will be pre-images of natural projections from the inverse limit of closed cubes tamely embeddedA \(d\) -dimensional cube \(B\subset \mathbb {R}^d\)  is tamely embedded in the interior of \(X\)  if \(B\)  is the image of the unit ball of \(\mathbb {R}^d\)  under a continuous one-to-one map from \(\mathbb {R}^d\)  into \(X\) . in \(\mathbb {D}^4\) , where \(\mathbb {D}^4\)  denotes the four dimensional unit cube. Therefore, our rectangles will be of the form of closed cubes from \(\mathbb {D}^4\)  times a Cantor set. Despite this change in the setting, basic tools from [1]} still hold true and we will be able to perform an adaptation of the Denjoy-Rees technique through inverse limits. Our basic starting point is to construct a Cantor set \(K\subset \mathcal {P}_{\infty } \subset \mathbb {D}_{\infty }\)  of positive Haar measure. Then the orbits of \(K\)  are “blown-up” similarly as in Rees' construction and the Denjoy-Rees technique will in addition allow us to completely control the orbits of points from \(K\)  so that no measure escapes to an uncontrolled parts of \(\mathcal {P}_{\infty }\setminus K\) . Therefore we will be able to control invariant measures on \(\mathcal {P}_{\infty }\) , subsequently also metric entropy as required and use the variational principle to determine explicit values of topological entropy as well.
